FractionalGeometry-Chap2 - Chapter 2 Iterated function...

Chapter 2Iterated function systems:fractals as limitsOften a Frst exposure to fractals consists of googling “fractal geometry,” Fndinga program to generate fractals, selecting some example from a preset menu, andclicking the RUN button. The image drifts into being, dot after dot dancingacross the screen in furious pattern. If the rules generating this fractal can beviewed with the program, the Frst question is “How do these rules make thatpicture?” The answer to that question is the Frst topic of this chapter, if anI±S program had the highest google rank.Before starting this relatively long analysis, a moment’s consideration of aself-similar fractal suggests a general direction for the proof. Look at the fractalshown in the left of ±ig. 2.1. This shape, the Sierpinski gasket, is one of thesimplest examples of a self-similar fractal. The middle image, a magniFcationof a small portion of the left side, is indistinguishable from the left. The rightis a magniFcation of a portion of the middle. In our minds, this process can becontinued forever.±igure 2.1: Successive magniFcations of portions of the gasket.The reappearance, under increasingly high magniFcation, of copies of thewhole shape, suggests two things. ±irst, some sort of limiting process is in-volved here. Second, it is not the limiting process familiar from calculus, whereunder magniFcation a smooth curve approaches its tangent line. Something dif-ferent is happening here: the level of complexity remains about constant undermagniFcation. So we must determine what sort of limit produces fractals.25

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26CHAPTER 2. ITERATED FUNCTION SYSTEMS2.1Iterated function system formalismBased on work of Mandelbrot [102] and Hutchinson [81], and popularized byBarnsley [5], iterated function systems are a formalism for generating fractalsand for compressing images. First we recall some background on transformationsof the plane.In the plane, a general linear transformation plus translation can be writtenasTbxyB=brcos(θ)−ssin(ϕ)rsin(θ)scos(ϕ)BbxyB+befB(2.1)That is, any real 2×2 matrix can be expressed as the 2×2 in eq (2.1). (SeeProb. 2.1.1.)Letd(,) denote the Euclidean distance. A transformationTis ad-contractionwith contraction factort, 0≤t <1, if for all points (x1,y1) and (x22),dpTbx1y1B,Tbx2y2BP≤t·dpbx1y1B,bx2y2BP,(2.2)and for any numbers < t,dpTbx1y1Bbx2y2BP> s·dpbx1y1B,bx2y2BPfor at least one pair of points (x11) and (x22).For example, ifT(x,y) = (x/2,y/2), thendpTbx1y1Bbx2y2BP=r±x12−x22²2+±y12−y22²2=12dpbx1y1B,bx2y2BPfor any pair of points. Consequently, thisTis a contraction with contractionfactor 1/2. A slightly more di±cult example is given in Prob. 2.1.2.

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